A τ-tilting approach to dissections of polygons

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A τ -tilting approach to dissections of polygons

Vincent Pilaud, Pierre-Guy Plamondon, Salvatore Stella

To cite this version:

Vincent Pilaud, Pierre-Guy Plamondon, Salvatore Stella. A τ -tilting approach to dissections of polygons. Symmetry, Integrability and Geometry: Methods and Applications, 2018, 14 (045), �10.3842/SIGMA.2018.045�. �hal-02343595�

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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 045, 8 pages

A τ -Tilting Approach to Dissections of Polygons

Vincent PILAUD a_{, Pierre-Guy PLAMONDON} b _{and Salvatore STELLA} c a _{CNRS & LIX, ´}_{Ecole Polytechnique, Palaiseau, France}

E-mail: vincent.pilaud@lix.polytechnique.fr

URL: _{http://www.lix.polytechnique.fr/~pilaud/}

b _{Laboratoire de Math´ematiques d’Orsay, Universit´e Paris-Sud,}

CNRS, Universit´e Paris-Saclay, France E-mail: pierre-guy.plamondon@math.u-psud.fr

URL: _{https://www.math.u-psud.fr/~plamondon/}

c _{University of Haifa, Israel}

E-mail: stella@math.haifa.ac.il

URL: _{http://math.haifa.ac.il/~stella/}

Received February 26, 2018, in final form May 10, 2018; Published online May 14, 2018 https://doi.org/10.3842/SIGMA.2018.045

Abstract. We show that any accordion complex associated to a dissection of a convex polygon is isomorphic to the support τ -tilting simplicial complex of an explicit finite dimen-sional algebra. To this end, we prove a property of some induced subcomplexes of support τ -tilting simplicial complexes of finite dimensional algebras.

Key words: dissections of polygons; accordion complexes; τ -tilting theory; representations of finite dimensional algebras; g-vectors

2010 Mathematics Subject Classification: 16G10; 16G20; 05E10

1 Introduction

The theory of cluster algebras gave rise to several interpretations of associahedra [14, 15, 16]. Fig. 1 shows two such interpretations for the rank 3 associahedron: as the exchange graph of triangulations of a hexagon and as the exchange graph of support τ -tilting modules over the cluster tilted algebra whose quiver with relations is as depicted. This follows from results in the setting of the “additive categorification of cluster algebras” that was initiated in [3,5].

F. Chapoton observed a similar isomorphism between the exchange graph of certain dissec-tions of a heptagon and that of support τ -tilting modules over the path algebra of the quiver

1 2 3, subject to the relation βα = 0. Fig.2shows these two exchange graphs, which can be found in [6, Fig. 7] and in [1, Example 6.4].

The purpose of this note is to show that this isomorphism is an avatar of a more general result in the theory of τ -tilting modules. Any basic finite dimensional algebra Λ gives rise to an exchange graph on support τ -tilting modules. This exchange graph is the dual graph of the support τ -tilting complex [1] (see Section 2). Let _{e1, . . . , en} be a complete set of primitive

pairwise orthogonal idempotents of Λ. Let J be a non-empty subset of [n] and eJ := Pj∈Jej.

The following result forms the algebraic core of the paper (see Section 2 for definitions and Section 3for the proof).

Theorem 1.1. The support τ -tilting complex of eJΛeJ is isomorphic to the subcomplex of the

support τ -tilting complex of Λ induced by the support τ -tilting modules whose g-vectors’ coordi-nates vanish outside of J.

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2 2 V. Pilaud, P.-G. Plamondon and S. StellaV. Pilaud, P.-G. Plamondon and S. Stella D_◦= 2_{1 ⊕}3_{2 ⊕}1₃ 3⊕ 3_{2 ⊕}1₃ 2_{1 ⊕ 1 ⊕}1₃ 2_{1 ⊕}3_{2 ⊕ 2} 3⊕ 3₂ 3_{2 ⊕ 2} 3⊕ 1₃ 1⊕ 1₃ 2_{1 ⊕ 1} 2_{1 ⊕ 2} 3 1 2 ∅ Q = 1 2 3

Figure 1. The exchange graph on triangulations of a hexagon (left) and the exchange graph on support τ -tilting modules of the quiver with relations Q (right).

D_◦= 1⊕ 2_{1 ⊕}3₂ 2⊕ 2_{1 ⊕}3₂ 1⊕ 3_{2 ⊕ 3} 1⊕ 2₁ 2⊕ 2₁ 2⊕ 3₂ 3⊕ 3₂ 1⊕ 3 2 3 1 ∅ Q(D_◦) = 1 2 3

Figure 2. The D◦-accordion complex of the dissection D◦of Fig.3(left) and the 2-term silting complex

of the quiver Q(D◦) (right).

With this algebraic result, we can explain the isomorphism of Fig.2, and extend it to dis-sections of any polygon. Any reference dissection of a polygon gives rise to an exchange graph on certain dissections. This exchange graph is the dual graph of the accordion complex studied in [6,10,12] (see Section4).

Theorem 1.2. Any accordion complex is isomorphic to the support τ -tilting simplicial complex of an explicit finite dimensional algebra. Thus, the corresponding exchange graphs are isomor-phic.

Note that this statement could be seen as a consequence of a more general result [13, Proposi-tion 2.44]. However, the latter combines several isomorphisms (from dissecProposi-tions, via non-kissing paths, through non-attracting chords, to support τ -tilting modules) and the proof is quite intri-cate. In the present paper, we will easily deduce Theorem1.2from Theorem1.1and from the

2 V. Pilaud, P.-G. Plamondon and S. Stella

D_◦= 2_{1 ⊕}3_{2 ⊕}1₃ 3⊕ 3_{2 ⊕}1₃ 2_{1 ⊕ 1 ⊕}1₃ 2_{1 ⊕}3_{2 ⊕ 2} 3⊕ 3₂ 3_{2 ⊕ 2} 3⊕ 1₃ 1⊕ 1₃ 2_{1 ⊕ 1} 2_{1 ⊕ 2} 3 1 2 ∅ Q = 1 2 3

D_◦= 1⊕ 2_{1 ⊕}3₂ 2⊕ 2_{1 ⊕}3₂ 1⊕ 3_{2 ⊕ 3} 1⊕ 2₁ 2⊕ 2₁ 2⊕ 3₂ 3⊕ 3₂ 1⊕ 3 2 3 1 ∅ Q(D_◦) = 1 2 3

Figure 2. The D_◦-accordion complex of the dissection D_◦of Fig.3(left) and the 2-term silting complex of the quiver Q(D◦) (right).

With this algebraic result, we can explain the isomorphism of Fig. 2, and extend it to dis-sections of any polygon. Any reference dissection of a polygon gives rise to an exchange graph on certain dissections. This exchange graph is the dual graph of the accordion complex studied in [6,10,12] (see Section4).

Note that this statement could be seen as a consequence of a more general result [13, Proposi-tion 2.44]. However, the latter combines several isomorphisms (from dissecProposi-tions, via non-kissing paths, through non-attracting chords, to support τ -tilting modules) and the proof is quite intri-cate. In the present paper, we will easily deduce Theorem1.2from Theorem1.1and from the Figure 2. The D_◦-accordion complex of the dissection D_◦of Fig.3(left) and the 2-term silting complex of the quiver Q(D◦) (right).

With this algebraic result, we can explain the isomorphism of Fig. 2, and extend it to dis-sections of any polygon. Any reference dissection of a polygon gives rise to an exchange graph on certain dissections. This exchange graph is the dual graph of the accordion complex studied in [6,10,12] (see Section4).

Note that this statement could be seen as a consequence of a more general result [13, Proposi-tion 2.44]. However, the latter combines several isomorphisms (from dissecProposi-tions, via non-kissing paths, through non-attracting chords, to support τ -tilting modules) and the proof is quite intri-cate. In the present paper, we will easily deduce Theorem 1.2 from Theorem1.1 and from the known case of triangulations of polygons (see Section 4).

The explicit finite dimensional algebras that appear in Theorem1.2had previously appeared in [7, 8], where gentle algebras are associated to dissections of any surface without puncture. Applying Theorem1.1to these algebras, one could obtain an analogue of the accordion complex for these surfaces.

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A τ -Tilting Approach to Dissections of Polygons 3

2 Recollections on τ -tilting theory

The theory of τ -tilting modules was introduced in [1], and we mainly follow this source. Let k be an algebraically closed field, let Λ be a basic finite-dimensional k-algebra, and let{e1, . . . , en}

be a complete set of pairwise orthogonal idempotents in Λ. Denote by mod Λ the category of finite-dimensional right Λ-modules, and by proj Λ its full subcategory of projective modules. We denote by τ the Auslander–Reiten translation of mod Λ (see, for instance, [2, Chapter IV]). For any Λ-module M , we denote by _{|M| the number of pairwise non-isomorphic direct summands} appearing in any decomposition of M into indecomposable modules.

2.1 Support τ -tilting pairs

Following [1, Definition 0.1], we say that a Λ-module M is τ-rigid if HomΛ(M, τ M ) = 0;

τ-tilting if it is τ-rigid and |M| = |Λ|;

support τ-tilting if there exists an idempotent e of Λ such that e is in the annihilator of M and M is a τ -tilting Λ/(e)-module.

Support τ -tilting modules always exist: Λ itself and the zero module are two examples.

It is useful to keep track of the idempotents in the annihilator of a support τ -tilting module. For this reason, we will follow [1, Definition 0.3] and call a pair (M, P ), with M _{∈ mod Λ and} P ∈ proj Λ, a

τ-rigid pair if M is τ-rigid and HomΛ(P, M ) = 0;

support τ-tilting pair if it is a τ-rigid pair and |M| + |P | = |Λ|;

almost complete support τ-tilting pair if it is a τ-rigid pair and |M| + |P | = |Λ| − 1. We will say that the pair (M, P ) is basic if both M and P are basic Λ-modules. We define direct sums of pairs componentwise.

One of the main theorems of [1] is the following.

Theorem 2.1 ([1, Theorem 0.4]). A basic almost complete support τ -tilting pair is a direct summand of exactly two basic support τ -tilting pairs.

Definition 2.2. The support τ -tilting complex of Λ is the simplicial complex sτ_{C(Λ) whose} vertices are the isomorphism classes of indecomposable τ -rigid pairs and whose faces are sets of τ -rigid pairs whose direct sum is rigid. The exchange graph sτ -tilt(Λ) is the dual graph of sτ_{C(Λ), i.e., the graph whose vertices are isomorphism classes of basic support τ-tilting pairs,} and where two vertices are joined by an edge whenever the corresponding support τ -tilting pairs differ by exactly one direct summand.

2.2 2-term silting objects

The study of support τ -tilting pairs turns out to be equivalent to that of another class of objects: 2-term silting objects [1, Section 3]. Let Kb_{(proj Λ) be the homotopy category of}

bounded complexes of projective Λ-modules. Let 2-cpx(Λ) be the full subcategory of Kb_{(proj Λ)}

consisting of 2-term objects, that is, complexes P = · · · → Pm+1 → Pm→ Pm−1 → · · ·

such that Pm is zero unless m∈ {0, 1}. We will write P1→ P0 to denote the complex

· · · → 0 → P1 → P0 → 0 → · · ·

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it is rigid, and |P| = |Λ|.

This is a special case of a more general definition of silting objects, see [11]. Examples of 2-term silting objects include 0_{→ Λ and Λ → 0.}

Definition 2.3. The 2-term silting complex of Λ is the simplicial complexSC(Λ) whose vertices are isomorphism classes of indecomposable rigid 2-term objects in Kb_{(proj Λ) and whose faces}

are sets of such objects whose direct sum is rigid. The exchange graph 2-silt(Λ) is the dual graph of SC(Λ), i.e., the graph whose vertices are isomorphism classes of basic 2-term silting objects in Kb_{(proj Λ), and where two vertices are joined by an edge whenever the corresponding objects}

differ by exactly one direct summand. For any Λ-module M , denote by PM

1 → P0M a minimal projective presentation of M .

Theorem 2.4 ([1, Theorem 3.2]). The map (M, P ) _{7→ P}M

1 → P0M ⊕ (P → 0) induces an

isomorphism of simplicial complexes sτ_{C(Λ) ∼}= SC(Λ), and thus of exchange graphs sτ-tilt(Λ) ∼

= 2-silt(Λ).

2.3 The g-vector of a 2-term object

The results of this note rely on the following definition.

Definition 2.5. Let_{P be a 2-term object in 2-cpx(Λ). The g-vector of P, denoted by g(P), is} the class of_{P in the Grothendieck group K}0 Kb(proj Λ).

We will usually denote g-vectors as integer vectors by using the basis of the abelian group K0 Kb(proj Λ)

given by the classes of the indecomposable projective modules e1Λ, . . . , enΛ

concentrated in degree 0. Thus, if P is the 2-term object M i∈[n] (eiΛ)⊕bi −→ M i∈[n] (eiΛ)⊕ai,

then its g-vector is g(P) = (ai− bi)i∈[n].

In contrast to arbitrary 2-term objects, rigid 2-term objects are determined by their g-vector in the following sense.

Theorem 2.6 ([9, Sections 2.3 and 2.4]). LetP and Q be two rigid 2-term objects. (i) If g(_{P) = g(Q), then P and Q are isomorphic.}

(ii) The object _{P is isomorphic to an object of the form (P}1 → P0)⊕ Q idQ

→ Q, where P1

and P0 do not have non-zero direct summands in common.

Note that Qid→ QQ is isomorphic to zero in Kb_{(proj Λ).}

3 Algebraic result

We use the same notations as in the previous section. In particular, Λ is a basic finite-dimensional k-algebra with complete set of pairwise orthogonal idempotents {e1, . . . , en}.

Let J be a subset of [n]. We will study 2-term objects that only involve the indecomposable projective modules ejΛ with j∈ J.

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Definition 3.1. Let 2-cpx_J(Λ) be the full subcategory of cpx(Λ) whose objects are the 2-term objects P1 → P0 such that all the indecomposable direct summands of P1 and P0 have the

form ejΛ with j∈ J.

Our main interest will lie in the rigid objects in 2-cpx_J(Λ).

Definition 3.2. LetSCJ(Λ) be the subcomplex ofSC(Λ) induced by J, that is, the subcomplex

whose vertices are rigid objects in 2-cpx_J(Λ). Let 2-siltJ(Λ) be the dual graph of SCJ(Λ). Its

vertices are isomorphism classes of basic objects _{P in 2-cpx}_J(Λ) satisfying P is rigid;

if P0 ∈ 2-cpxJ(Λ) andP ⊕ P0 is rigid, thenP0 is a direct sum of direct summands of P.

Two vertices are joined by an edge whenever the corresponding objects differ by exactly one indecomposable direct summand.

In other words, the faces ofSCJ(Λ) correspond to basic rigid objects whose g-vectors have zero

coefficients in entries corresponding to elements not in J. In this sense,_SCJ(Λ) is a

representa-tion-theoretic analogue of the accordion complex [6,10,12] (see Theorem4.3). This is the main motivation for the introduction of this object.

Let eJ := Pj∈Jej, and consider the algebra eJΛeJ. Observe that eJΛeJ is isomorphic to

EndΛ(ΛeJ). This has the following consequence. Let projJ(Λ) be the full subcategory of proj(Λ)

whose objects are isomorphic to direct sums of the indecomposable modules ejΛ, with j ∈ J.

Lemma 3.3. The k-linear categories proj_J(Λ) and proj(eJΛeJ) are equivalent. In particular,

the categories Kb_(proj

J(Λ)) and Kb(proj(eJΛeJ)) are equivalent.

This lemma immediately implies the following statement.

Theorem 3.4. The simplicial complexesSCJ(Λ) andSC(eJΛeJ) are isomorphic. In particular,

their dual graphs 2-siltJ(Λ) and 2-silt(eJΛeJ) are isomorphic.

Corollary 3.5. The simplicial complex _SCJ(Λ) is a pseudomanifold of dimension |J| − 1. In

particular, its dual graph 2-siltJ(Λ) is|J|-regular.

4 Application: accordion complexes of dissections

Let P be a convex polygon. We call diagonals of P the segments connecting two non-consecutive vertices of P. A dissection of P is a set D of non-crossing diagonals. It dissects the polygon into cells. We denote by Q(D) the quiver with relations whose vertices are the diagonals of D, whose arrows connect any two counterclockwise consecutive edges of a cell of D, and whose relations are given by triples of counterclockwise consecutive edges of a cell of D. See Fig. 3for an example.

We now consider 2m points on the unit circle alternately colored black and white, and let P_◦ (resp. P_•) denote the convex hull of the white (resp. black) points. We fix an arbitrary reference dissection D_◦of P_◦. A diagonal δ_•of P_•is a D_◦-accordion diagonal if it crosses either none or two consecutive edges of any cell of D_◦. In other words, the diagonals of D_◦ crossed by δ_• together with the two boundary edges of P_◦ crossed by δ_• form an accordion. A D_◦-accordion dissection is a set of non-crossing D_◦-accordion diagonals. See Fig.3for an example. We call D_◦-accordion complex the simplicial complex_AC(D_◦) of D_◦-accordion dissections. This complex was studied in recent works of F. Chapoton [6], A. Garver and T. McConville [10], and T. Manneville and V. Pilaud [12].

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Figure 3. A dissection D◦ with its quiver Q(D◦) (left), a D◦-accordion diagonal (middle) and a D◦

-accordion dissection (right).

For a diagonal δ_◦ of D_◦ and a D_◦-accordion diagonal δ_• intersecting δ_◦, we consider the three edges (including δ_◦) crossed by δ_• in the two cells of D_◦ containing δ_◦. We define ε δ_◦ ∈ D◦| δ•

to be 1, _{−1, or 0 depending on whether these three edges form a Z, a Z , or a VI. The g-vector} of δ_• with respect to D_◦ is the vector g(D_◦_{| δ}_•)_{∈ R}D◦ _{whose δ}

◦-coordinate is ε δ◦ ∈ D◦| δ•.

Example 4.1. When the reference dissection D_◦ is a triangulation of P_◦, any diagonal of P_• is a D_◦-accordion diagonal. The D_◦-accordion complex is thus an n-dimensional associahedron (of type A), where n = m− 3. As explained in [5], the D_◦-accordion complex is isomorphic to the 2-term silting complex of the quiver Q(D_◦) of the triangulation D_◦. The isomorphism sends a diagonal of P_• to the 2-term silting object with the same g-vector. See Fig. 1 for an illustration.

With the notations we introduced, we can now restate Theorem1.2 more precisely.

Theorem 4.2. For any reference dissection D_◦, the D_◦-accordion complex is isomorphic to the 2-term silting complex of the quiver Q(D_◦).

One possible approach to Theorem4.2would be to provide an explicit bijective map between D_◦-accordion diagonals and 2-term silting objects for Q(D_◦). Such a map is easy to guess using g-vectors, but the proof that it is actually a bijection and that it preserves compatibility is intricate. This approach was developed in the more general context of non-kissing complexes of gentle quivers with relations in [13, Propostion 2.44]. In this note, we use an alternative simpler strategy to obtain Theorem 4.2 by using Theorem 1.1and understanding accordion complexes as certain subcomplexes of the associahedron.

For that, consider two nested dissections D_◦ _{⊂ D}0_◦. Observe that any D_◦-accordion diagonal is a D0_◦-accordion diagonal. Conversely a D0_◦-accordion diagonal δ_• is a D_◦-accordion diagonal if and only if it does not cross any diagonal δ0_◦ of D0_◦r D_◦ as a Z or a Z , that is if and only if the δ_◦0-coordinate of its g-vector g(D0_◦_{| δ}_•) vanishes for any δ0_◦ _{∈ D}0_◦r D_◦. This observation shows the following statement.

Theorem 4.3 ([12, Section 4.2]). For any two nested dissections D_◦ _{⊂ D}0_◦, the accordion complex_AC(D_◦) is isomorphic to the subcomplex of_AC(D0_◦) induced by D0_◦-accordion diagonals δ_• whose g-vectors g(D0_◦| δ•) lie in the coordinate subspace spanned by elements in D◦.

In order to prove Theorem 4.2 we now turn to associative algebras. Let Q = (Q, I) be any gentle quiver with relations [4] and J be any subset of vertices of Q. We call shortcut quiver the quiver with relations Q_J = (QJ, IJ) whose vertices are the elements of J, whose arrows are the

paths in Q with endpoints in J but no internal vertex in J, and whose relations are inherited from those of Q. Then the quotient kQJ/IJ of the path algebra kQJ is gentle and is isomorphic

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Example 4.4. Quivers of dissections are shortcut quivers: if D_◦ _{⊂ D}0_◦, then Q(D_◦) = Q(D0_◦)D◦.

In particular, for any dissection D_◦, the quiver Q(D_◦) is a shortcut quiver of the quiver with relations of a cluster tilted algebra of type A.

The following statement is an application of Theorem1.1to gentle algebras.

Theorem 4.5. For any gentle quiver with relations Q and any subset J of vertices of Q, the 2-term silting complex _SC(Q_J) for the shortcut quiver Q_J is isomorphic to the subcomplex of the 2-term silting complex SC(Q) induced by 2-term silting objects whose g-vectors lie in the coordinate subspace spanned by vertices in J.

Combining Theorems 4.3 and 4.5 together with Example 4.1, we obtain Theorem 4.2 (and Theorem1.2).

5 Concluding remarks

Remark 5.1. There is a geometric interpretation of the common phenomenon described in Theorems4.3and4.5. For a D_◦-accordion dissection D_•, denote byR_≥0g(D_◦_{| D}_•) the polyhedral cone generated by the set of g-vectors g(D_◦| D•) :={g(D◦| δ•)| δ• ∈ D•}. The collection Fg(D◦)

of cones R_≥0g(D_◦_{| D}_•) for all D_◦-accordion dissections D_• is a complete simplicial fan called g-vector fan of D_◦ [12]. The crucial feature of this fan is that no coordinate hyperplane meets the interior of any of its maximal cones. This is often referred to as the sign-coherence property of g-vectors. It implies that for any two nested dissections D_◦⊂ D0

◦, the section ofFg(D0◦) with

the coordinate subspace RD◦ _{is a subfan of} _Fg_(D0

◦). The content of Theorem 4.3 is that this

subfan is the g-vector fanFg_(D

◦). A similar statement holds for Theorem 4.5.

Remark 5.2. In the theory of cluster algebras, a standard operation consists of freezing a subset of the initial cluster. This corresponds to taking a section of the d-vector fan by a coordinate subspace. To the best of our knowledge, the same operation on the g-vector fan studied in this note was not considered before in the literature.

Remark 5.3. The connection between representation theory and accordion complexes was already considered by A. Garver and T. McConville in [10, Section 8]. However, their approach deals with c-vectors and simple-minded collections while our approach deals with g-vectors and silting objects.

Acknowledgements

We are grateful to F. Chapoton for pointing out to us the isomorphism between the two graphs of Fig. 2 which gave us the motivation for the present note. We also thank R. Schiffler for his comments on a previous version. Finally, we are grateful to an anonymous referee for helpful suggestions on the presentation of this note. The first two authors are partially supported by the French ANR grant SC3A (15 CE40 0004 01). The last author is supported by the ISF grant 1144/16.

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